Intertwining operators of the quantum Teichmüller space

Abstract

In arXiv:0707.2151 the authors introduced the theory of local representations of the quantum Teichmüller space ${\mathcal{T}}_S^q$ ($q$ being a fixed primitive $N$-th root of $(-1{)}^{N+1}$) and they studied the behaviour of the intertwining operators in this theory. One of the main results [Theorem 20, arXiv:0707.2151] was the possibility to select one distinguished operator (up to scalar multiplication) for every choice of a surface $S$, ideal triangulations $\lambda ,{\lambda }^{\prime }$ and isomorphic local representations $\rho ,{\rho }^{\prime }$, requiring that the whole family of operators verifies certain Fusion and Composition properties. By analyzing the constructions of arXiv:0707.2151, we found a difficulty that we eventually fix by a slightly weaker (but actually optimal) selection procedure. In fact, for every choice of a surface $S$, ideal triangulations $\lambda ,{\lambda }^{\prime }$ and isomorphic local representations $\rho ,{\rho }^{\prime }$, we select a finite set of intertwining operators, naturally endowed with a structure of affine space over $H_1(S;{\mathbb{Z}}_N)$ (${\mathbb{Z}}_N$ is the cyclic group of order $N$), in such a way that the whole family of operators verifies augmented Fusion and Composition properties, which incorporate the explicit behavior of the ${\mathbb{Z}}_N$-actions with respect to such properties. Moreover, this family is minimal among the collections of operators verifying the "weak" Fusion and Composition rules (in practice the ones considered in arXiv:0707.2151). In addition, we adapt the derivation of the invariants for pseudo-Anosov diffeomorphisms and their hyperbolic mapping tori made in arXiv:0707.2151 and arXiv:math/0407086 by using our distinguished family of intertwining operators.

85 pages, 9 figures

References in corpus (1)

• Local representations of the quantum Teichmuller space